Incoherent Effect of Space Charge and Electron Cloud

Trapping in, or scattering off, resonances driven by space charge (SC) or electron cloud (EC) in conjunction with synchrotron motion can explain observations of slow beam loss and emittance growth, which are often accompanied by changes in the longitudinal beam profile. This talk will review recent progress in understanding and modeling the underlying mechanisms, highlight the differences and similarities between space charge and electron cloud, and discuss simulation results in the light of experimental observations, e.g., at GSI, CERN and BNL. SC & EC INCOHERENT EFFECTS Space harge ncoherent ffects The term “incoherent effects” of space charge in a 2D beam normally refers to the incoherent tuneshift of each particle in a beam [1]. Coherent space effects in transverse plane are more related to the collective beam response to the beam perturbations [2, 3]. The interplay of the coherent tuneshift with lattice driven resonances or structure resonances is essential for the correct identification of the tunes where the resonant effect will take place [4]. These studies are made mainly for 2D beams. The request of long term storage of high intensity bunches brought to the attention in circular accelerator the full 3D problem. The beam dynamics of a bunch is approximated by partially decoupling the dynamics of the transverse-longitudinal planes: the synchrotron motion is considered, in first approximation, independent. As the transverse-longitudinal frequency ratio is typically large, Qx/Qz > 500, parametric resonances are excluded. The only remaining effect of the synchrotron motion on the particles in a bunch is to advance them longitudinally and via space charge induce a transverse tune modulation at a frequency twice the synchrotron frequency. In the CERN benchmarking experiment [5] this mechanism was tested under controlled experimental conditions. It was found that the beam response and beam loss are consistent with the numerical modeling. The underlying mechanism for this beam response relies on the space charge transverse tune modulation for inducing a periodic resonance crossing. In this beam dynamics regime trapping/scattering of beam particles into the resonance creates a complex diffusive dynamics which becomes evident only after many synchrotron oscillations. Only the particles, which cross the resonance are subjected to trapping/scattering and this condition of “resonance crossing” depends on the initial particle invariants ǫx, ǫy, ǫz , the space charge tuneshift ∆Qx,sc, and the working point (Qx0, Qy0). In Ref. [5] it is shown that the maximum amplitude a particle can ∗ [email protected] reach depends on the distance from the resonance approximately as∼ 1/(Qx−Qx,res). This dependence creates two regimes: a beam loss regime for tunes located in the proximity of the resonance (above), and a neighboring emittance growth regime (no beam loss). In Ref. [6] the role of the transverse tune dependence induced by space charge is discussed for a Gaussian stationary bunched beam. The fraction of particles to be trapped/scattered is estimated as ∆N/N ∼ (Qx −Qx,res)/∆Qx,sc. As only particles with large synchrotron amplitude will span the full space charge tune-spread and therefore may reach a large transverse amplitude, the beam loss will shorten the bunch length [7]. Recently also the role of chromaticity in the 3D high intensity bunched beams has been explored and it is found that it enhances beam loss bringing the numerical results closer to the experimental findings [8]. Electron Cloud Incoherent Effects The presence of the electron cloud in proton machines has been always associated with the creation of instabilities [9, 10]. The interaction of localized electrons with proton beams is very complex in terms of formation and dynamics: when a proton bunch passes through a localized electron cloud it causes a pinch of the electron cloud itself [11, 12]. The idea that the pinched electron cloud is also responsible for the creation of incoherent effects on the proton beam has been around for several years. At the ICFA-HB2004 workshop, the analogy with space charge induced trapping phenomena was brought into the discussion. The essential key suggesting a similarity with space charge is the correlation of the amount of pinch with the extent of the bunch that has passed through the EC. This correlation creates a dependence of the pinch experienced by a bunch particle and its longitudinal position inside the bunch at the time of passage through the EC [13]. In this dynamics the electrons are the weak beam as it is subjected to large variations in density, which however may “resonantly” feed back on the strong main beam. For a bunch longer than the EC extension, the EC pinch occurs several times for the same electrons [11] according to the bunch charge density and sizes. The possibility of trapping/scattering induced by pinched EC is shown in [14]. There a simplified model of EC is used by assuming the EC kick modeled by an EC beam of density linearly growing from the bunch head to the tail. This model showed that a small emittance growth can be created similarly to what happens with space charge. Clearly the prediction capability of such a model is based on the modeling of the pinched EC. Simulations in fact show that the EC pinch progresses as the bunch goes through the EC and exhibits a complicated time dependent EC morphology with “rings” [12]. A previous attempt to model the effect of such rings C I E THYM02 Proceedings of EPAC08, Genoa, Italy 05 Beam Dynamics and Electromagnetic Fields 2942 D03 High Intensity Incoherent Instabilities, Space Charge, Halos, Cooling is reported in Ref. [15] where a one dimensional model is studied. We here extend the EC modeling to EC rings and compare its effect on the bunch dynamics with that induced by SC. SC & EC INCOHERENT EFFECTS: DIFFERENCES AND SIMILARITIES The main difference between the SC and EC is that the SC force scales with the beam energy as 1/γ making the high intensity effects negligible at high energy, while EC forces still remain relevant for the beam dynamics. Another difference is in the shape of the Coulomb force much more nonlinear than for EC. Both SC and EC create a transverse amplitude dependent detuning which is a function of the beam distribution for the space charge, and of the EC pinch morphology for the electron clouds. The EC is often localized in specific regions of the ring creating a distribution of kicks on the strong beam the strength of which depends on the particle position within the bunch. The pinching of the EC causes always two effects: 1) the correlation of EC intensity of the pinch with the position along the bunch; 2) systematic resonances of even order. In the bunch reference frame the structure of the EC density assumes quite a complex form during the pinch process, which makes its effect on the main beam dynamics particularly difficult to asses in long term storage. Note that SC may create systematic resonances too of a strength consistent with the harmonics of the lattice optics. In this respect both SC and EC create structure resonances. In terms of incoherent effects, the main difference arises from the complex dependence of the amplitude dependent detuning, which characterizes the efficiency of trapping/scattering regimes [6]. In order to compare SC and EC incoherent effects we model the beam dynamics in a constant focusing lattice and for the sake of simplicity we consider two special frozen models, one for the SC, and another one for the EC as follows: • We consider a stationary bunched beam where the particle distribution forms a 3D Gaussian distribution ρ(x, y, z) = exp[−(x2 + y2)/(2σ)− z/(2σz)] from which the SC can be found analytically [6]; • Based on simulation results of the EC pinch [12, 16] we construct a simplified frozen model formed by 3 EC rings that are created along the bunch at the locations of the 3 pinches zp = −1σz, 0.3σz, 1.5σz . Each EC ring has a radial thickness of 1σ of the beam, and its radial position is R(z) = 3.33 × (z − zp) for z > zp. As simplifying ansatz the model assumes an electron charge conservation inside each EC ring and that the EC electric field is well described by a “cylindrical sheet” approximating the EC rings allowing then a straightforward calculation of the electric field. This model extends the previously studied one dimensional sheet model presented in Ref. [15]. Comparison of EC & SC Incoherent Effects in We first consider an example with tunes as in the SPS and study the transport of a high intensity axi-symmetric bunched beam in presence of a lattice nonlinearity excited via a single octupole (similarly to what was done experimentally in the CERN-PS experiment [5]). The single 0.4 0.6 0.8 1 1.2 1.4 1.6 26.1 26.15 26.2 26.25 26.3 εx / εx0 εy / εy0